On generalization of Lipschitz groups and spin groups

TL;DR

This paper explores new Lie groups that preserve specific subspaces within geometric algebras, extending the concepts of Lipschitz and spin groups, and analyzes their Lie algebras for broader mathematical understanding.

Contribution

It introduces generalized Lie groups related to geometric algebra subspaces, expanding the classical Lipschitz and spin groups, and studies their Lie algebra structures.

Findings

Some Lie groups generalize Lipschitz and spin groups.

These groups coincide with classical groups in small dimensions.

Lie algebras of these groups are characterized.

Abstract

This paper presents some new Lie groups preserving fixed subspaces of geometric algebras (or Clifford algebras) under the twisted adjoint representation. We consider the cases of subspaces of fixed grades and subspaces determined by the grade involution and the reversion. Some of the considered Lie groups can be interpreted as generalizations of Lipschitz groups and spin groups. The Lipschitz groups and the spin groups are subgroups of these Lie groups and coincide with them in the cases of small dimensions. We study the corresponding Lie algebras.